Near Minimax Line Spectral Estimation

• Published in: IEEE transactions on information theory Vol. 61; no. 1; pp. 499 - 512
• Main Authors: Gongguo Tang, Bhaskar, Badri Narayan, Recht, Benjamin
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2015; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximate support recovery; Atomic clocks; atomic norm; compressive sensing; Estimating techniques; Experiments; Frequency estimation; infinite dictionary; line spectral estimation; minimax rate; Noise measurement; Noise reduction; Resolution; Semidefinite programming; sparsity; Spectral analysis; Spectrum analysis; stable recovery; superresolution; Approximate support recovery; stable recovery; infinite dictionary; line spectral estimation; superresolution; minimax rate; sparsity; atomic norm; compressive sensing
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2368122

This paper establishes a nearly optimal algorithm for denoising a mixture of sinusoids from noisy equispaced samples. We derive our algorithm by viewing line spectral estimation as a sparse recovery problem with a continuous, infinite dictionary. We show how to compute the estimator via semidefinite programming and provide guarantees on its mean-squared error rate. We derive a complementary minimax lower bound on this estimation rate, demonstrating that our approach nearly achieves the best possible estimation error. Furthermore, we establish bounds on how well our estimator localizes the frequencies in the signal, showing that the localization error tends to zero as the number of samples grows. We verify our theoretical results in an array of numerical experiments, demonstrating that the semidefinite programming approach outperforms three classical spectral estimation techniques.